Abstract and Applied Analysis
Volume 2011 (2011), Article ID 962313, 18 pages
http://dx.doi.org/10.1155/2011/962313
Research Article

Algebraic Properties of Toeplitz Operators on the Polydisk

Bo Zhang,1 Yanyue Shi,2 and Yufeng Lu1

1School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China
2School of Mathematical Sciences, Ocean University of China, Qingdao 266100, China

Received 3 August 2011; Revised 29 September 2011; Accepted 2 October 2011

Academic Editor: Alberto d'Onofrio

Copyright © 2011 Bo Zhang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We discuss some algebraic properties of Toeplitz operators on the Bergman space of the polydisk Dn. Firstly, we introduce Toeplitz operators with quasihomogeneous symbols and property (P). Secondly, we study commutativity of certain quasihomogeneous Toeplitz operators and commutators of diagonal Toeplitz operators. Thirdly, we discuss finite rank semicommutators and commutators of Toeplitz operators with quasihomogeneous symbols. Finally, we solve the finite rank product problem for Toeplitz operators on the polydisk.